Question

How do you simplify `f(theta)=-2csc(theta/4)-cot(theta/2)-3cos(theta/4)` to trigonometric functions of a unit `theta`?

Answer 1

`f(theta)=+-2/(sqrt((1+-sqrt((1+cos(theta))/2))/2))+-sqrt((1+-sqrt((1+cos(theta))/2))/2)/(sqrt((1-cos(theta))/2))+-3sqrt((1+-sqrt((1+cos(theta))/2))/2)`

Explanation:

Assuming the question is asking you to get rid of the fractions, it is important to know your half-angle formulas:

• `sin(u/2)=+-sqrt((1-cos(u))/2`
• `cos(u/2)=+-sqrt((1+cos(u))/2)`
• `tan(u/2)=(1-cos(u))/sin(u)`

Step 1. Change `csc` and `cot` in terms of `sin` and `cos`

`f(theta)=-2/sin(theta/4)-cos(theta/2)/sin(theta/2)-3cos(theta/4)`

Step 2. Express each term so that it has `theta"/"2`

`f(theta)=-2/sin((theta"/"2)/2)-cos(theta/2)/sin(theta/2)-3cos((theta"/"2)/2)`

Step 3. Replace each `sin` and `cos` with half-angle formulas

`f(theta)=+-2/(sqrt((1-cos(theta/2))/2))+-sqrt((1+cos(theta/2))/2)/(sqrt((1-cos(theta))/2))+-3sqrt((1+cos(theta/2))/2)`

Step 4. Replace each `theta"/"2` term with it its half-angle formula

`f(theta)=+-2/(sqrt((1+-sqrt((1+cos(theta))/2))/2))+-sqrt((1+-sqrt((1+cos(theta))/2))/2)/(sqrt((1-cos(theta))/2))+-3sqrt((1+-sqrt((1+cos(theta))/2))/2)`

I'll leave simplifying this as homework!